Geodesic
All-at-once preconditioning in PDE-constrained optimization
Kybernetika, Tome 46 (2010) no. 2, pp. 341-360.

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The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques.
Classification : 49J20, 49M25, 65F08, 65F10, 65F50, 65K10, 65N22, 76D07
Keywords: optimal control; preconditioning; partial differential equations 
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     author = {Rees, Tyrone and Stoll, Martin and Wathen, Andy},
     title = {All-at-once preconditioning in {PDE-constrained} optimization},
     journal = {Kybernetika},
     pages = {341--360},
     publisher = {mathdoc},
     volume = {46},
     number = {2},
     year = {2010},
     mrnumber = {2663605},
     zbl = {1195.65083},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010__46_2_a8/}
}
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Rees, Tyrone; Stoll, Martin; Wathen, Andy. All-at-once preconditioning in PDE-constrained optimization. Kybernetika, Tome 46 (2010) no. 2, pp. 341-360. http://geodesic.mathdoc.fr/item/KYB_2010__46_2_a8/